Comparison between explicit and implicit discretization strategies for a dissipative thermal environment

TL;DR

The paper addresses how to efficiently simulate open quantum systems coupled to dissipative baths by contrasting explicit bath discretization (ML-MCTDH) with implicit bath discretization (TTN-HEOM). It leverages a common tensor-network framework (TENSO) to implement both approaches and uses bath-correlation-function decompositions, notably for Drude–Lorentz and Brownian environments, to benchmark performance. The key finding is that TTN-HEOM, which encodes bath memory through a small number of bexcitonic auxiliary modes, achieves numerically exact dynamics with far fewer DoFs and far lower time cost than explicit discretization, particularly for dissipative dynamics with rapidly decaying BCFs; explicit methods can approximate short-time behavior but require many bath modes to capture long-time thermalization. The results across two-level and seven-site FMO models demonstrate substantial practical gains in both computational efficiency (time 40–70× faster; memory 2–3×) and scalability, informing best practices for simulating realistic dissipative quantum systems in chemistry and biophysics.

Abstract

We investigate strategies for simulating open quantum systems coupled to dissipative baths by comparing explicit wave function-based discretization [via multi-layer multi-configuration time-dependent Hartree (ML-MCTDH)] and the implicit density matrix-based master equation method [via tree tensor network hierarchical equations of motion (TTN-HEOM)]. For dissipative baths characterized by exponentially decaying bath correlation functions, the implicit discretization approach of HEOM -- rooted in bath correlation function decompositions -- proves significantly more efficient than explicit discretization of the bath into discrete harmonic modes. Explicit methods, like ML-MCTDH, require extensive mode discretization to approximate continuum baths, leading to computational bottlenecks. Case studies for two-level systems and a Fenna--Matthews--Olson complex model highlight TTN-HEOM's superiority in capturing dissipative dynamics with relaxations with a minimal number of auxiliary modes, while the explicit methods are as exact as the HEOM in pure dephasing regimes. This comparison is enabled by the TENSO package, which has both ML-MCTDH and TTN-HEOM implemented using the same computational structure and propagation strategy.

Figures (12)

• Figure 1: Discretized spectral density developed from the (a) logarithmic, (b) equalized, and (c) BSDO discretization strategies. The spectral density $\mathcal{J}(\omega)$ is the Drude--Lorentz (DL) one with reorganization energy $\lambda_\text{D} = 200~\mathrm{cm}^{-1}$ and characteristic frequency $\gamma_\text{D} = 100~\mathrm{cm}^{-1}$, and temperature is set to be $T = 300~\text{K}$. The cutoff frequency of the discretization is set at $1000~\mathrm{cm}^{-1}$. For the logarithmic discretization, the minimum frequency $\omega_\text{min}$ is set to be $0.01~\mathrm{cm}^{-1}$. The number of discretized DoFs is set to be $J=120,\ 240$ for each panel. The implicit discretization using the Padé $[2/3]$ approximant with $K=4$ (in yellow solid) and the exact target $\mathcal{J}(\omega)(1+n(\omega, T))$ (in black dashed) are also plotted for comparison.
• Figure 2: Errors in capturing the pure dephasing dynamics of a two-level system using TTN-HEOM and ML-MCTDH. The two-level system is prepared in a superposition of the two states $\lvert\psi(0)\rangle = (\lvert0\rangle + \lvert1\rangle)/\sqrt{2}$ and coupled to a DL bath at $300$ K. Here the reorganization energy is $\lambda_\text{D} = 200~\mathrm{cm}^{-1}$ and the characteristic frequency $\gamma_\text{D} = 100~\mathrm{cm}^{-1}$. ML-MCTDH is used to simulate the discretized dynamics with logarithmic (log.), equalized (equal.), and BSDO discretization strategies with $J=240$ and cutoff frequency $1000~\mathrm{cm}^{-1}$. The TTN-HEOM is used to simulate the implicit discretization using the Padé $[2/3]$ approximant with $K=4$. The solid lines are the error of the coherence $\left\lvert {[\rho_{\text{S}}]_{01}(t)} \right\rvert$, while the dashed lines are the error of the population $\left\lvert {[\rho_{\text{S}}]_{00}(t)} \right\rvert$. The number in the bracket for each label indicates the number of discretized DoFs used in the simulation.
• Figure 3: Dissipative dynamics of a two-level system with relaxation using different methods. The two-level system is prepared in $\lvert\psi(0)\rangle = \lvert e\rangle = (\lvert0\rangle + \lvert1\rangle)/\sqrt{2}$ and is coupled with a DL bath at $300$ K. Here the reorganization energy $\lambda_\text{D} = 200~\mathrm{cm}^{-1}$ and the characteristic frequency $\gamma_\text{D} = 100~\mathrm{cm}^{-1}$. ML-MCTDH are used to simulate the discretized dynamics with logarithmic, equalized, and BSDO discretization strategies with $J=240$ (solid lines) and cutoff frequency $1000~\mathrm{cm}^{-1}$. The TTN-HEOM is used to simulate the implicit discretization using the Padé $[2/3]$ approximant with $K=4$. The upper panels show the population in the excited state $[\rho_\text{S}]_{ee}(t)$ of $H_\text{S}$, and the lower panels show the coherence purity $\mathrm{Tr}\rho^2_\text{S}(t)$. The dotted lines are the results from the ML-MCTDH with logarithmic, equalized and BSDO discretization strategies with $J=240$ and the correction Eq. \ref{['eq:ch7-eta']} applied. The number in the bracket for each label indicates the number of discretized DoFs used in the simulation.
• Figure 4: The same dynamics as Fig. \ref{['fig:toy2']} but with a larger cutoff frequency $3000~\mathrm{cm}^{-1}$ and the number of discretized DoFs is set to be $J=120$ and $J=240$ for the ML-MCTDH computations.
• Figure 5: Population and purity dynamics in FMO using ML-MCTDH with different discretization strategies (dashed lines: equivalent reorganization energy discretization strategy; dash-dotted lines: logarithmic discretization ; dotted lines: BSDO discretization) and TTN-HEOM with implicit discretization based on Padé approximant (solid lines).